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2. 1(b) by the relevant estimated standard deviation. Again, the two sets of intervals are similar, indicating that the approximation is probably a reasonable one, at least for this particular combination of sample size and effect size. 28 (estimated) standard deviation units. 24 population standard deviation units. Unlike the raw intervals, the two sets of standardized intervals differ slightly in their midpoints, because the raw midpoints are divided by slightly different quantities (the constant population standard deviation in the case of the z procedure, and the variable sample standard deviation in the case of the t procedure).

10) as the definition of a contrast on means. That is, ψ1 = c1′ µ is a compact way of defining the contrast ψ1 . We will occasionally use vector or matrix notation to summarize simple operations involving the calculation of sums of products. In the main, however, we will use vector (or matrix) notation to refer to individual vectors (or matrices), not to provide a basis for the mathematical operations of matrix algebra. Familiarity with matrix algebra will not be assumed. The scale of contrast coefficients The magnitude of the population value of a contrast depends on the vector of population means and the pattern and scale of the contrast coefficients.

For example, the coefficient vector c1b pattern as c1′ : the two sets of coefficients are proportional and therefore ′ µ must be twice as large as the perfectly correlated. The value of ψ1b = c1b 36 Analysis of Variance via Confidence Intervals value of ψ1 , simply because of the difference between c1 and c1b in the scale (magnitude) of the contrast coefficients. Any interpretation of the magnitude of a contrast must take into account the scaling of the contrast coefficients. Mean difference contrasts A mean difference contrast compares the (weighted or unweighted) average of a subset of means with the average of a different (nonoverlapping) subset.